Dual-Mode Multi-modulus Algorithms for Blind Equalization of QAM Signals

Wen si-yuan Shandong Economic University

Jinan, China Wsy1116@Sohu.com

Liu Feng Jessie88_ly@163.com

AbstractA dual-mode multi-modulus algorithm (DM- MMA) and a stop-and-go dual-mode multi-modulus algorithm (SAG-DMMMA) for blind equalization of high-order quadrature amplitude modulation (QAM) signals are proposed. Simulation results show the proposed blind equalization algorithms have faster convergence speed and smaller steady-state mean square error, compared with the recently introduced multi-modulus algorithm.

Keywords- blind equalization; dual-mode; multi-modulus algorithm; stop-and-go.

I. INTRODUCTION The increasing demand for digital communications needs

high-speed data transmission over band-limited channels. Hence, the channels are subject to intersymbol interference (ISI). Channel equalization is one of the techniques to mitigate the effects of ISI. Blind equalization which does not require any known training sequence has been an active area for several decades.

The constant modulus algorithm (CMA)[1]has become a favorite of practitioners due to its LMS-like complexity and desirable robustness properties. But the CMA converges independently of carrier recovery, and the output constellation after convergence has a phase rotation. Thus a rotator has to be added at the output of the equalizer, which increases the complexity of implementation of the receiver in steady-state operation. In order to improve the performance of the CMA, a multi-modulus algorithm (MMA) has been proposed [2,3,4]. The MMA provides reliable initial convergence without the need of a rotator in steady-state.

The MMA cost function is [2]

(1)

2I

2I

2R

2RMMA nyny EJ

)()(

where E[] indicates statistical expectation. yR(n) and yI(n) are the real and imaginary parts of the equalizer output y(n), respectively. R and I are computed as

, 2

4

I2

4

)n(aE)n(aE

)n(aE)n(aE

I

I

R

RR

(2)

where a(n)=aR(n) + j aI(n) is the transmitted QAM data symbol. The corresponding MMA tap updating algorithm is

(3) )()()()1( nnenn XWW Where is the step-size parameter and the asterisk

denotes complex conjugation. The equalizer complex tap weight-vector and input-vector are respectively defined as

W(n)=[w0(n), w1(n),, wL-1(n)]T and X(n)=[x(n), x(n-1),, x(n-L+1)]T , where L is the length of the equalizer tap weights and the notation superscript T stands for transpose of vector. The error function e(n) can be expressed as

)()()()()( nynyjnynyne 2III2RRR

. (4) A particular problem of the MMA, however, is that it still

suffers from a low convergence when applies to the higher order QAM. The dual-mode blind equalization algorithms are therefore designed to speed up the convergence rate [5,6,7]. These algorithms use the CMA at the first mode and then switch to the second mode, so they suffer from the same pitfall as the CMA, i.e. they cant correct the phase rotation at the output of the equalizer. In this paper, we propose a new dual-mode MMA to effectively improve the equalizers convergence performance and recover the phase rotation simultaneously.

Although the dual-mode algorithm switches in two modes, it never stops adjusting the equalizer tap weights even when the adjustment is in the wrong direction. If we can tell whether a particular adjustment is correct or not, we may improve the convergence behavior by making only the right adjustment but bypassing those wrong ones. Such a concept has been applied to blind equalization and is termed stop-and-go [8,9,10]. In this paper, we develop a stop-and-go dual-mode MMA for blind equalizers.

The paper is organized as follows. In Section 2 we derive and analyze the proposed algorithms. Computer simulations are presented in Section 3, and the concluding remarks are contained in Section 4.

II. THE STOP-AND-GO DUAL-MODE ALGORITHM

A. The dual-mode multi-modulus algorithmSuppose D k (k=1,2,) denotes the union of the square

regions Dk (as shown in Fig. 1 using the 32-QAM signal as an example) enclosing data points of the QAM data constellation. In these regions, we define a new cost function:

22

21 )n(a)n(y)n(a)n(yEJ~ q,IIp,RR (5)

where i nanyminargp iRRi ,,,)()( , 21 (6)

k nanyminargq kIIk ,,,)()( , 21 . (7)

_____________________________________ 978-1-4244-5265-1/10/$26.00 2010 IEEE

By computing the gradient of (5) with respect to wR(n) and wI(n) separately, and replacing the value of statistical expectation by the instantaneous value, we obtain:

)n()n(y)n(a)n(y

)n()n(y)n(a)n(yJ~

IIq,II

RRp,RRR

X

XW

sgn

sgn (8)

)n()n(y)n(a)n(y

)n()n(y)n(a)n(yJ~

RIq,II

IRp,RRI

X

XW

sgn

sgn (9)

Figure 1. Square regions for the DM-MMA of a 32-QAM signal.

Then the corresponding tap updating algorithm is (10) )()(~)()1( nnenn XWW The error signal in (10) is expressed as

kIR Dnynejnene )( ),(~)(~)(~ (11) where )n(y)n(a)n(y)n(e~ Rp,RRR sgn (12) )n(y)n(a)n(y)n(e~ Iq,III sgn . (13) We refer to the above algorithm as the constellation

region based MMA (CR-MMA). From (3), (4), (10), (11) and the preceding discussion, we

may express the dual-mode MMA (DM-MMA) as

kDnynnenn )( ),()()()1( XWW (14)

kDnynnenn )( ),()(~)()1( XWW (15)

The DM-MMA operates as follows. Because of the distortion introduced by the channel during the transient stages, the output of the equalizer will be scattered in a very large area around the transmitted data point. Therefore, most of the equalizer outputs will not be in Dk and the MMA (i.e. (14)) will be used to adjust the tap weights most of the time. This provides a MMA-like initial convergence behavior for the DM-MMA. On the other hand, in the steady state, since the output of the equalizer will be very close to

the transmitted data point, the CR-MMA (i.e. (15)) will be used to adjust the tap weights most of the time. When yR(n)aR(n) and yI(n)aI(n), the error signal in (11) will have a very small value, which provides a good steady state behavior for the DM-MMA.

B. The stop-and-go dual-mode multi-modulus algorithmAlthough the DM-MMA correctly decide whether the

equalizer is in a transient state or a steady state, it does not tell whether a particular adjustment is correct or not. The stop-and-go method can solve this problem by using a simple flag. The flag suggests go if the error signal is sufficiently reliable for adaptation, and suggests stop otherwise [8, 9]. In the following, we present a stop-and-go DM-MMA (SAG-DMMMA) for blind equalization. The SAG-DMMMA can be described by the following equations:

,2,1,)( ),()()()()()()1(

kDnynnenfjnenfnn

k

IIRR XWW

(16)

,2,1,)( ),()(~)()(~)()()1(

kDnynnenfjnenfnn

k

IIRR XWW

(17) where

)(~)(,

)( ~)(,)(

nesgnne sgnif

nesgnne sgnif nf

RR

RR

R0

1 (18)

)(~)(,

)( ~)(,)(

nesgnne sgnif

nesgnne sgnif nf

II

II

I0

1. (19)

III. SIMULATIONS We have demonstrated the performance of the DM-

MMA and SAG-DMMMA using computer simulations. The data sequences for simulations were generated according to the model of Fig. 2. In this model, a T-spaced (where T is the symbol period) symbol sequence {a(n)} is transmitted through the channel, which is the overall complex baseband equivalent impulse response of the transmitter filter, unknown channel and receiver filter. The channel output is corrupted by additive white Gaussian noise (AWGN). The received signal r(n) is interpolated by a factor of two and filtered by a T/2-spaced feedforward filter. After that, the output is decimated by a factor of two to get the T-spaced sequence {y(n)} for the posterior decision device.

The channel used in simulations was taken from [8]. A decision-feedback equalizer with a 5-tap T/2-spaced feedforward filter and a 3-tap feedback filter was used. The equalizer was initialized so that the center taps were set to one and the other taps were set to zero. The signal to noise ratio (SNR) was fixed at 20 dB in all simulations.

In Fig. 3 we show that for the different d values (the width of each square is 2d as depicted in Fig. 1), there has a

great difference in the mean square error (MSE) of the proposed algorithm. From this figure, it can be seen that the performance of the algorithm deteriorates when d is increased beyond 0.85, and for d=1 the algorithm fails to converge. Note that when d=1, the square regions touch each other, and this case is very similar to the CR-MMA equalization scheme.

Next, to examine the performance of the proposed algorithm, we compared the DM-MMA and SAG-DMMMA with the CMA and the MMA for the 32-QAM, 128-QAM and 256-QAM data constellations. The step-size is set to 0.000012 and 0.00004 for the CMA and the MMA with the 32-QAM data set, respectively. For the DM-MMA and SAG-DMMMA, d=0.7 and =0.002, 0.0001 and 0.00002 for the 32-QAM, 128-QAM and 256-QAM constellations, respectively. Fig. 4 shows the constellations after convergence. Fig. 4(a) shows for the CMA of a 32-QAM signal. It is clear that the equalized output constellation has an arbitrary phase rotation introduced by the channel, which has not been corrected. Fig. 4(b), 4(c) and 4(d) are the constellations of 32-QAM signals for the MMA, DM-MMA and the SAG-DMMMA, respectively. The phase rotation has been recovered for these algorithms and the DM-MMA and SAG-DMMMA can achieve better performance compared with the MMA. In addition, we can see from Fig. 4 that the symbols are more clearly distinguishable at the output of the SAG-DMMMA equalizers than those of the DM-MMA. Note that the CMA and the MMA did not succeed in opening the eye for the 128-QAM and 256-QAM signals, so we have not presented their data constellations.

Finally, we consider the convergence behavior and steady state MSE of these algorithms. Fig. 5 shows the ensemble-averaged MSE, obtained from 100 Monte Carlo runs. From the results, we can see that the SAG-DMMMA has the fastest convergence rate and the smallest steady state MSE among all the algorithms.

Figure 2. Baseband communication system model with T/2- spaced

receiver.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

-25

-20

-15

-10

-5

0

Iterations

MS

E(d

B)

MSE of DM-MMA for different d values

d=1

d=0

d=0.3

d=0.1

d=0.5

d=0.7

d=0.85

Figure 3. Learning curves for DM-MMA for a 16-QAM system. For d=0,

the corresponding step-size are =0.00002; for d=1, =0.001, and others =0.002.

(a)CMA for 32QAM (b) MMA for 32QAM

(c) DM-MMA for 32QAM (d) SAG-DMMMA for 32QAM

(e) DM-MMA for 128QAM (f) SAG-DMMMA for 128QAM

(g) DM-MMA for 256QAM (h) SAG-DMMMA for 256QAM

Figure 4. Output of the equalizers after 4000 iterations.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

Iterations

MS

E(d

B)

CMA for 32QAM

MMA for 32QAM

DM-MMA for 32QAM

SAG-DMMMA for 32QAM

DM-MMA for 128QAM

SAG-DMMMA for 128QAM

DM-MMA for 256QAM

SAG-DMMMA for 256QAM

Figure 5. Comparison curves of ensemble averaged MSE.

IV. CONCLUSIONS In this paper we have introduced two new multi-modulus

algorithms for blind equalization and have analyzed their

performance. The DM-MMA and the SAG-DMMMA outperform the MMA and offer practical alternatives to blind equalization of higher-order QAM signals.

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